The result of Mehler can also be linked to probability. For this, the variables should be rescaled as x → x/√2, y → y/√2, so as to change from the 'physicist's' Hermite polynomials H(.) (with weight function exp(−x2)) to "probabilist's" Hermite polynomials He(.) (with weight function exp(−x2/2)). Then, E becomes
The left-hand side here is p(x,y)/p(x)p(y) where p(x,y) is the bivariate Gaussian probability density function for variables x,y having zero means and unit variances:
and p(x), p(y) are the corresponding probability densities of x and y (both standard normal).
There follows the usually quoted form of the result (Kibble 1945)[8]
This expansion is most easily derived by using the two-dimensional Fourier transform of p(x,y), which is
This may be expanded as
The Inverse Fourier transform then immediately yields the above expansion formula.
This result can be extended to the multidimensional case.[8][9][10]
This is a continuous family of linear transforms generalizing the Fourier transform, such that, for α = π/2, it reduces to the standard Fourier transform, and for α = −π/2 to the inverse Fourier transform.
The Mehler formula, for ρ = exp(−iα), thus directly provides
The square root is defined such that the argument of the result lies in the interval [−π /2, π /2].
If α is an integer multiple of π, then the above cotangent and cosecant functions diverge. In the limit, the kernel goes to a Dirac delta function in the integrand, δ(x−y) or δ(x+y), for α an even or odd multiple of π, respectively. Since [f ] = f(−x), [f ] must be simply f(x) or f(−x) for α an even or odd multiple of π, respectively.
^Mehler, F. G. (1866), "Ueber die Entwicklung einer Function von beliebig vielen Variabeln nach Laplaceschen Functionen höherer Ordnung", Journal für die Reine und Angewandte Mathematik (in German) (66): 161–176, ISSN 0075-4102, ERAM 066.1720cj (cf. p 174, eqn (18) & p 173, eqn (13) )
^Pauli, W., Wave Mechanics: Volume 5 of Pauli Lectures on Physics (Dover Books on Physics, 2000) ISBN 0486414620 ; See section 44.
^The quadratic form in its exponent, up to a factor of −1/2, involves the simplest (unimodular, symmetric) symplectic matrix in Sp(2,R). That is,
where
so it preserves the symplectic metric,
^Horvathy, Peter (1979). "Extended Feynman Formula for Harmonic Oscillator". International Journal of Theoretical Physics. 18 (4): 245-250. Bibcode:1979IJTP...18..245H. doi:10.1007/BF00671761. S2CID 117363885.
^Wolf, Kurt B. (1979), Integral Transforms in Science and Engineering, Springer ([1] and [2]); see section 7.5.10.
^Celeghini, Enrico; Gadella, Manuel; del Olmo, Mariano A. (2021). "Hermite Functions and Fourier Series". Symmetry. 13 (5): 853. arXiv:2007.10406. Bibcode:2021Symm...13..853C. doi:10.3390/sym13050853.
^ abKibble, W. F. (1945), "An extension of a theorem of Mehler's on Hermite polynomials", Proc. Cambridge Philos. Soc., 41 (1): 12–15, Bibcode:1945PCPS...41...12K, doi:10.1017/S0305004100022313, MR 0012728, S2CID 121931906
^Slepian, David (1972), "On the symmetrized Kronecker power of a matrix and extensions of Mehler's formula for Hermite polynomials", SIAM Journal on Mathematical Analysis, 3 (4): 606–616, doi:10.1137/0503060, ISSN 0036-1410, MR 0315173
^Hörmander, Lars (1995). "Symplectic classification of quadratic forms, and general Mehler formulas". Mathematische Zeitschrift. 219: 413–449. doi:10.1007/BF02572374. S2CID 122233884.
^Wiener, N (1929), "Hermitian Polynomials and Fourier Analysis", Journal of Mathematics and Physics8: 70–73.
^Condon, E. U. (1937). "Immersion of the Fourier transform in a continuous group of functional transformations", Proc. Natl. Acad. Sci. USA23, 158–164. online
Nicole Berline, Ezra Getzler, and Michèle Vergne (2013). Heat Kernels and Dirac Operators, (Springer: Grundlehren Text Editions) Paperback ISBN 3540200622
Louck, J. D. (1981). "Extension of the Kibble-Slepian formula for Hermite polynomials using boson operator methods". Advances in Applied Mathematics. 2 (3): 239–249. doi:10.1016/0196-8858(81)90005-1.
H. M. Srivastava and J. P. Singhal (1972). "Some extensions of the Mehler formula", Proc. Amer. Math. Soc.31: 135–141. (online)